Turbulent Rayleigh–Bénard convection (RBC) [1]}, [2]}, [3]},
which occurrs in a fluid layer due to a temperature difference at its bottom and top surfaces, is
the paradigmatic model system to study thermally driven turbulence.
Usually it is considered within the Oberbeck–Boussinesq (OB) approximation,
in which all fluid properties are assumed to be constant, apart from the density in the buoyancy term of the momentum equation, where it is assumed to be linearly dependent on the temperature.
However, in many cases in nature and technology,
the density of many fluids is strongly non-linear and even non-monotonic with the temperature, which significantly influences the flow patterns and the heat transport properties in the system.
The most famous and relevant example is water, for which the density
is maximal at \(T_m\approx 4^\circ \rm {C}\) .
This density maximum has a drastic
influence on many natural phenomena like the freezing of lakes and estuaries, and the survival of fauna in shallow waters in winter [4]}, [5]}, [6]}, [7]}.
However, also here RBC can be used again as paradigmatic model system, but now
correspondingly RBC of cold water near \(T_m\) .
This is an example of
so-called penetrative convection [8]}, [9]}, where
convection in a thermally unstable layer penetrates into the adjacent stable layers. Penetrative convection is
not only important for water around \(4^\circ \) C and freezing lakes etc., but also in many astrophysical settings,
for example, in the tachocline of the Sun [10]}, and possibly in the liquid core of the Earth [11]} and in the Jupiter's atmosphere [12]}.
